Optimal. Leaf size=88 \[ -\frac {5 x}{128}-\frac {5 \cosh (a+b x) \sinh (a+b x)}{128 b}-\frac {5 \cosh ^3(a+b x) \sinh (a+b x)}{192 b}-\frac {\cosh ^5(a+b x) \sinh (a+b x)}{48 b}+\frac {\cosh ^7(a+b x) \sinh (a+b x)}{8 b} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2648, 2715, 8}
\begin {gather*} \frac {\sinh (a+b x) \cosh ^7(a+b x)}{8 b}-\frac {\sinh (a+b x) \cosh ^5(a+b x)}{48 b}-\frac {5 \sinh (a+b x) \cosh ^3(a+b x)}{192 b}-\frac {5 \sinh (a+b x) \cosh (a+b x)}{128 b}-\frac {5 x}{128} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2648
Rule 2715
Rubi steps
\begin {align*} \int \cosh ^6(a+b x) \sinh ^2(a+b x) \, dx &=\frac {\cosh ^7(a+b x) \sinh (a+b x)}{8 b}-\frac {1}{8} \int \cosh ^6(a+b x) \, dx\\ &=-\frac {\cosh ^5(a+b x) \sinh (a+b x)}{48 b}+\frac {\cosh ^7(a+b x) \sinh (a+b x)}{8 b}-\frac {5}{48} \int \cosh ^4(a+b x) \, dx\\ &=-\frac {5 \cosh ^3(a+b x) \sinh (a+b x)}{192 b}-\frac {\cosh ^5(a+b x) \sinh (a+b x)}{48 b}+\frac {\cosh ^7(a+b x) \sinh (a+b x)}{8 b}-\frac {5}{64} \int \cosh ^2(a+b x) \, dx\\ &=-\frac {5 \cosh (a+b x) \sinh (a+b x)}{128 b}-\frac {5 \cosh ^3(a+b x) \sinh (a+b x)}{192 b}-\frac {\cosh ^5(a+b x) \sinh (a+b x)}{48 b}+\frac {\cosh ^7(a+b x) \sinh (a+b x)}{8 b}-\frac {5 \int 1 \, dx}{128}\\ &=-\frac {5 x}{128}-\frac {5 \cosh (a+b x) \sinh (a+b x)}{128 b}-\frac {5 \cosh ^3(a+b x) \sinh (a+b x)}{192 b}-\frac {\cosh ^5(a+b x) \sinh (a+b x)}{48 b}+\frac {\cosh ^7(a+b x) \sinh (a+b x)}{8 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.07, size = 52, normalized size = 0.59 \begin {gather*} \frac {-120 b x-48 \sinh (2 (a+b x))+24 \sinh (4 (a+b x))+16 \sinh (6 (a+b x))+3 \sinh (8 (a+b x))}{3072 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 1.81, size = 61, normalized size = 0.69
method | result | size |
default | \(-\frac {5 x}{128}-\frac {\sinh \left (2 b x +2 a \right )}{64 b}+\frac {\sinh \left (4 b x +4 a \right )}{128 b}+\frac {\sinh \left (6 b x +6 a \right )}{192 b}+\frac {\sinh \left (8 b x +8 a \right )}{1024 b}\) | \(61\) |
risch | \(-\frac {5 x}{128}+\frac {{\mathrm e}^{8 b x +8 a}}{2048 b}+\frac {{\mathrm e}^{6 b x +6 a}}{384 b}+\frac {{\mathrm e}^{4 b x +4 a}}{256 b}-\frac {{\mathrm e}^{2 b x +2 a}}{128 b}+\frac {{\mathrm e}^{-2 b x -2 a}}{128 b}-\frac {{\mathrm e}^{-4 b x -4 a}}{256 b}-\frac {{\mathrm e}^{-6 b x -6 a}}{384 b}-\frac {{\mathrm e}^{-8 b x -8 a}}{2048 b}\) | \(117\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.27, size = 110, normalized size = 1.25 \begin {gather*} \frac {{\left (16 \, e^{\left (-2 \, b x - 2 \, a\right )} + 24 \, e^{\left (-4 \, b x - 4 \, a\right )} - 48 \, e^{\left (-6 \, b x - 6 \, a\right )} + 3\right )} e^{\left (8 \, b x + 8 \, a\right )}}{6144 \, b} - \frac {5 \, {\left (b x + a\right )}}{128 \, b} + \frac {48 \, e^{\left (-2 \, b x - 2 \, a\right )} - 24 \, e^{\left (-4 \, b x - 4 \, a\right )} - 16 \, e^{\left (-6 \, b x - 6 \, a\right )} - 3 \, e^{\left (-8 \, b x - 8 \, a\right )}}{6144 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.38, size = 138, normalized size = 1.57 \begin {gather*} \frac {3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{7} + 3 \, {\left (7 \, \cosh \left (b x + a\right )^{3} + 4 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{5} + {\left (21 \, \cosh \left (b x + a\right )^{5} + 40 \, \cosh \left (b x + a\right )^{3} + 12 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} - 15 \, b x + 3 \, {\left (\cosh \left (b x + a\right )^{7} + 4 \, \cosh \left (b x + a\right )^{5} + 4 \, \cosh \left (b x + a\right )^{3} - 4 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{384 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 189 vs.
\(2 (80) = 160\).
time = 0.94, size = 189, normalized size = 2.15 \begin {gather*} \begin {cases} - \frac {5 x \sinh ^{8}{\left (a + b x \right )}}{128} + \frac {5 x \sinh ^{6}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{32} - \frac {15 x \sinh ^{4}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{64} + \frac {5 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{6}{\left (a + b x \right )}}{32} - \frac {5 x \cosh ^{8}{\left (a + b x \right )}}{128} + \frac {5 \sinh ^{7}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{128 b} - \frac {55 \sinh ^{5}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{384 b} + \frac {73 \sinh ^{3}{\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{384 b} + \frac {5 \sinh {\left (a + b x \right )} \cosh ^{7}{\left (a + b x \right )}}{128 b} & \text {for}\: b \neq 0 \\x \sinh ^{2}{\left (a \right )} \cosh ^{6}{\left (a \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.39, size = 116, normalized size = 1.32 \begin {gather*} -\frac {5}{128} \, x + \frac {e^{\left (8 \, b x + 8 \, a\right )}}{2048 \, b} + \frac {e^{\left (6 \, b x + 6 \, a\right )}}{384 \, b} + \frac {e^{\left (4 \, b x + 4 \, a\right )}}{256 \, b} - \frac {e^{\left (2 \, b x + 2 \, a\right )}}{128 \, b} + \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{128 \, b} - \frac {e^{\left (-4 \, b x - 4 \, a\right )}}{256 \, b} - \frac {e^{\left (-6 \, b x - 6 \, a\right )}}{384 \, b} - \frac {e^{\left (-8 \, b x - 8 \, a\right )}}{2048 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.68, size = 53, normalized size = 0.60 \begin {gather*} \frac {\frac {\mathrm {sinh}\left (4\,a+4\,b\,x\right )}{128}-\frac {\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{64}+\frac {\mathrm {sinh}\left (6\,a+6\,b\,x\right )}{192}+\frac {\mathrm {sinh}\left (8\,a+8\,b\,x\right )}{1024}}{b}-\frac {5\,x}{128} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________